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![Problem 1: Exchange economy
Consider an exchange economy with two consumers, A and B, and two
goods, X and Y.
Consumer A has an initial endowment TA = i >0 of good X, an initial
endowment yA = 0 of good Y, and preferences over consumption bundles
that can be represented by the Cobb-Douglas utility function
uA(TA: YA) = ",
where ra is the quantity of good X, yYA is the quantity of good Y, and
a € (0, 1) is a preference parameter.
Consumer B has an initial endowment ig = 0 of good X, an initial endow-
ment js = j > 0 of good Y, and preferences over consumption bundles
that can be represented by the Cobb-Douglas utility function
up(rB, YB):
B1-3
where rB is the quantity of good X, YB is the quantity of good Y, and
BE (0, 1) is a preference parameter.
(a) Find the set of Pareto optimal allocations in this economy. In appro-
priate diagrams, illustrate the set of Pareto optimal allocations when
(i) 3 = a, (ii) 3 > a, and (iii) 3 < a.
(b) Find the competitive equilibrium prices and allocations in this econ-
omy. Illustrate the competitive equilibrium in an appropriate diagram
for the case when B> a.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff81fc2f7-5a5f-4616-8df0-44da440fe28f%2F4aa24ca3-e97b-432c-9019-0e0aea577fe2%2F5ht3pw_processed.jpeg&w=3840&q=75)
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- Neha and Sonam consume only Coke and Pepsi in a two person and two goods exchange economy. Neha consumes these two goods in fixed proportions such that she consumes 2 bottles of coke with 1 bottle of Pepsi. Sonam's utility function is given by U = 4P + 3C . In the economy, there is a total of 100 bottles of Pepsi and 200 bottles of Coke. If Neha initially had 60 bottles of Pepsi and 80 bottles of coke, then in the equilibrium position, Sonam will consume: A. between 50 to 60 bottles of Pepsi B. between 52 to 60 bottles of Pepsi C. between 54 to 60 bottles of Pepsi D. between 56 to 60 bottles of Pepsi1) Suppose that Abel and Eden spend their incomes on two goods, food (F) and clothing (C).Abel’s preferences are represented by the utility function U(F,C)=10FC, while Eden’spreferences are represented by the utility function U(F,C)= 0.20F2C2A) With food on the horizontal axis and clothing on the vertical axis, identify on a graph theset of points that give Abel the same level of utility as the bundle (10, 5). Do the same forEden on a separate graph.B) On the same two graphs, identify the set of bundles that give Abel and Eden the samelevel of utility as the bundle (15, 8).C) Do you think Abel and Eden have the same preferences or different preferences? Explain 2. Graphically show the effect of an increase in price of Coca Cola on the demand of PepsiCola 3. Assume a budget line is drawn for two commodities: X on the x-axis and Y on the y-axis. Ifthe income of the consumer is 120 Birr, the y-intercept is 3, and the slope of the budget lineis -0.5 then determined the price of commodity…Please answer every part. 4. Consider an economy consisting of two individuals, Ann and Bob, and two goods, scotch and wine. Aun has 5 bottles of scoteh and 2 bottles of wine as her endowment, while Bob has 3 bottles of each. Suppose their preferences are described by the following utility functions uA(s, w) = sw and up(s, w) = s'u. Assume also that the prices of goods scotch and wine are represented by P,= 1 (scotch is the mumeraire), and P>0. a. Sketch the Edgeworth box of the economy with Ann at the lower left corner and Bob at the upper right corner; scotch on the horizontal axis, and wine on the vertical axis. Indicate the endowment point e in the box. b. Write the budget lines for Ann and Bob. e. Solve Ann's utility maximization problem. Expross Ann's optimal consumption bundle in terms of P. d. Solve Bob's utility maximization problem. Express Bob's optimal consumption bundle in terms of P. e. Define competitive equilibrium. Compute and plot the CE for this problem.
- Ginny likes to eat Bertie Bott’s Every Flavour Beans (b), and Chocolate Frogs (f). Herpreferences over these lollies are Cobb-Douglas, and can be presented by the utility functionu(b, f) = 1/5(b − 2)(f + 3). Each Every Flavour Bean costs 25 Knuts and each Chocolate Frogcosts 12 Knuts. Ginny has 314 Knuts to spend. Assume that beans and frogs are both perfectlydivisible goods.(a) Find Ginny’s optimal bundle.(b) Suppose that Ginny is mugged by Draco on her way to the lolly cart, so that by the timeshe is there, she is left only with 75 Knuts. What is her optimal bundle now?(c) Now, consider the general case, where Ginny’s income is m, the price of Every FlavourBeans is pb, and the price of Chocolate Frogs is given by pf . In other words, treat ChocolateFrogs as the numeraire. Find Ginny’s demand for each good in terms of m, pb and pf .(d) Find the good on which Ginny will always spend more than half of her income.(e) Find the own-price, cross-price, and income elasticities of good 2.Ginny likes to eat Bertie Bott’s Every Flavour Beans (b), and Chocolate Frogs (f). Herpreferences over these lollies are Cobb-Douglas, and can be presented by the utility functionu(b, f) = 1/5(b − 2)(f + 3). Each Every Flavour Bean costs 25 Knuts and each Chocolate Frogcosts 12 Knuts. Ginny has 314 Knuts to spend. Assume that beans and frogs are both perfectlydivisible goods.(a) Find Ginny’s optimal bundle.(b) Suppose that Ginny is mugged by Draco on her way to the lolly cart, so that by the timeshe is there, she is left only with 75 Knuts. What is her optimal bundle now?(c) Now, consider the general case, where Ginny’s income is m, the price of Every FlavourBeans is pb, and the price of Chocolate Frogs is given by pf . In other words, treat ChocolateFrogs as the numeraire. Find Ginny’s demand for each good in terms of m, pb and pf .(d) Find the good on which Ginny will always spend more than half of her income.(e) Find the own-price, cross-price, and income elasticities of good 2…Let the following table represents the total utility of a given consumer, in the cardinal utility approach. Q 1 2 3 4 5 6 7 TUX 8 14 18 20 20 18 16 TUY 6 10 13 15 16 16 14 MUX MUY MUX/PX MUY/PY Calculate the MUX and MUY and fill the table in the 4th and 5th rows. If the two products (X&Y) are free goods how many of X and Y should the consumer take to maximize utility? What is the maximum utility of X and Y if they are free? Let now price of X is 4 birr per unit and price of Y is 2 birr per unit. Calculate MUX/PX and MUY/PY and fill the 6th and 7th row. Assuming the consumer has any amount of money (enough budget) how many of X and Y should the consumer buy, to maximize utility? What is the total utility of X and Y? Let now price of X is 4 birr per unit and price of Y is 2 birr per unit and budget of the consumer for consumption of X and Y is 20 birr. Given budget constraint how many of X and Y should the consumer buy to maximize utility?
- Suppose that Abel and Eden spend their incomes on two goods, food (F) and clothing (C).Abel’s preferences are represented by the utility function U(F,C)=10FC, while Eden’spreferences are represented by the utility function U(F,C)= 0.20F2C2A) With food on the horizontal axis and clothing on the vertical axis, identify on a graph theset of points that give Abel the same level of utility as the bundle (10, 5). Do the same forEden on a separate graph.B) On the same two graphs, identify the set of bundles that give Abel and Eden the samelevel of utility as the bundle (15, 8).C) Do you think Abel and Eden have the same preferences or different preferences? Explain5. Sheila and Bruce are taking a canoe trip. Sheila brought 10 boxes of peanuts (x) and 15 bags of chips (y). Sheila's utility function is U*(x,y) = lnvx*+ Invy°. Bruce also brought 20 boxes of peanuts and 5 bags of chips. Bruce's utility function is UB(x.y) = min[x', y'I. a) Illustrate the endowment point and draw a sample set of indifference curves through the endowment point. b) If Sheila and Bruce trade what will be the pattern of mutually beneficial trade? c) If the terms of trade are the number of bags of chips (y) per box of peanuts (x) then what is the largest value that these terms can be for a mutually beneficial trade in this economy? d) Find one mutually beneficial trade where the terms of trade are 1 bag of chips (y) per 2 boxes of peanuts (x). Suppose that Sheila and Bruce set up two competitive markets for peanuts and chips. Below you will show that if the price of peanuts (x) is $1 and the price of chips (y) is $2 then the markets for both peanuts and chips will clear.…2) Two consumers R and S share an endowment (x, y) of goods X and Y. The consumers' utilities from consumption can be written as UR = aln(XR) + bln(YR), and Us=aln(xs) + bln(ys), where XR, XS, YR, and ys are the quantities which each consumes, and a and b are parameters, for which a + b =1. Write a report in which you a) Define the concept of the contract curve, and show that for these endowments, it will be the upward diagonal of the Edgeworth box, which shows all possible divisions of the endowment. b) Assume that in the initial endowment, XRE = XE and ysE=yE. R begins with the endowment of X and S begins with the endowment of Y. By setting up a constrained maximisation problem for each consumer, obtain their offer curves, and hence obtain the Walrasian equilibrium. c) State the two welfare theorems, and explain how they apply in this case.
- Let the following table represents the total utility of a given consumer, in the cardinal utility approach. Q 1 2 3 4 5 Tux 8 14 18 20 20 Tuy 6 10 13 15 16 Mux Muy Mux/px Muy/py A) Calculate the MUx and MUy and fill the table in the 4th and 5th rows. B) If the two products (X&Y) are free goods how many of X and Y should the cons consumer take to maximize utility? C) What is the maximum utility of X and Y if they are free?.If the initial distribution of two goods between two people is Pareto optimal, which of the following statements is TRUE? A. It is possible to reallocate the goods between the two people so as to increase the utility of both people. B. It is possible to reallocate the goods between the two people so as to increase the utility of one person without decreasing the utility of the other. C. It is possible to reallocate the goods between the two people so as to increase the utility of one person, but only at the expense of the other person. D. It is impossible to reallocate the goods between the two people so as to increase either person's utility. E. None of the aboveGood Y Use this figure of a feasible frontier and a set of an agent's indifference curves to answer the following three questions: A B C D E Good X shot 2024-... i Q Q Economists use the symbol ">" to mean "is preferred to". For example, if for Ash JK, it means that they prefer J to K. Which of the responses below correctly indicates this agent's preferences over the five labeled points on the feasible frontier, listed from most preferred to least preferred? OD C B E A OC B D A E OB A C D E OE D C B>A OA B C D E
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